A class of non-binary matroids with many binary miners

It is a well-known result of Tutte, A homotopy theorem for matroids, I, II, Trans. Amer. Math. Sec. 88 (1958) 144-174. that U-2,U-4 is the only non-binary matroid M such that, for every element e, both M\e and M/e are binary. Oxley generalized this result by characterizing the non-binary matroids M such that, for every element e of M, the deletion M\e or the contraction M/e is binary. We characterize those non-binary matroids M such that, for all elements e and f, at least two of M\e, f; M\e/f; M\e/f; and M\e, f are binary. (C) 1999 Elsevier Science B.V. All nights reserved.